Properties

Degree 1
Conductor $ 11 \cdot 547 $
Sign $0.906 + 0.422i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.485 − 0.873i)2-s + (0.309 − 0.951i)3-s + (−0.527 − 0.849i)4-s + (−0.262 − 0.964i)5-s + (−0.681 − 0.732i)6-s + (0.981 − 0.192i)7-s + (−0.998 + 0.0483i)8-s + (−0.809 − 0.587i)9-s + (−0.970 − 0.239i)10-s + (−0.970 + 0.239i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.998 − 0.0483i)15-s + (−0.443 + 0.896i)16-s + (0.0241 + 0.999i)17-s + (−0.906 + 0.421i)18-s + ⋯
L(s,χ)  = 1  + (0.485 − 0.873i)2-s + (0.309 − 0.951i)3-s + (−0.527 − 0.849i)4-s + (−0.262 − 0.964i)5-s + (−0.681 − 0.732i)6-s + (0.981 − 0.192i)7-s + (−0.998 + 0.0483i)8-s + (−0.809 − 0.587i)9-s + (−0.970 − 0.239i)10-s + (−0.970 + 0.239i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.998 − 0.0483i)15-s + (−0.443 + 0.896i)16-s + (0.0241 + 0.999i)17-s + (−0.906 + 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.906 + 0.422i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.906 + 0.422i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6017\)    =    \(11 \cdot 547\)
\( \varepsilon \)  =  $0.906 + 0.422i$
motivic weight  =  \(0\)
character  :  $\chi_{6017} (900, \cdot )$
Sato-Tate  :  $\mu(65)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 6017,\ (0:\ ),\ 0.906 + 0.422i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.8660172263 - 0.1920366313i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.8660172263 - 0.1920366313i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4210367956 - 1.047516315i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4210367956 - 1.047516315i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.1938992968114971196280044615, −17.633305197527342496028884476709, −16.57933622403593141371583786980, −16.46109096981025312196072720713, −15.42440940045596833500616120469, −15.07278800856705080113794015654, −14.51184818175834732239982770388, −13.97663047742809099063841557383, −13.61649344963205616894726720855, −12.22342471744085818004045273328, −11.68952072807965823256565871451, −11.246576455108075518796089671437, −10.23617847339512188165779233292, −9.62227963019587597179226699785, −8.92818684127602765081967892126, −8.16544868087498643404149460677, −7.43346304520207297262745616251, −7.15694919020060064826992850920, −5.98654084253930265346353164448, −5.30490998558048279685309082186, −4.8431633934089698463039089348, −3.95137276697316066344733183140, −3.46222371014093721700423995494, −2.65779210001324345375391192351, −1.8528198484021660822773905688, 0.18546054373875699046540690454, 0.950613067286231554517423310066, 1.848679578196444569487237431178, 2.1548373926638024762831608231, 3.277915566028028616630244040787, 3.99955104298967130820213517520, 4.74198586588287199645124615697, 5.48187321100284575562644159196, 5.932377619108045940727552928949, 7.19337803070201779921809494428, 7.73935540352013175030912718038, 8.50698287970627633272148291821, 8.97673044414064001420767423841, 9.80738229238869664126346327226, 10.694914984077240243606155867127, 11.34779480721274722624698886894, 12.11206155902585150703830592012, 12.349435293333240642337133776722, 13.14646116610472604873083399568, 13.56228135376981597930208812271, 14.39307895150077251836966258391, 14.85595036760485264483279160327, 15.50935615876999876778609749443, 16.63397768435814966845685059898, 17.42084248821005659141827276886

Graph of the $Z$-function along the critical line